| Density functions are of interest in many fields. In biodemographic studies, when one observes age-at-death for several cohorts of flies, time-variation may exist between cohort densities due to a cage effect. Given the cohort densities of age-at-death, one aims to estimate the underlying overall density. In microarray gene expression experiments, similar problems arise during normalization of probe-level data from different arrays. Conventional methods ignore this type of variation and hence lead to a non-representative overall density. We propose to view densities as functional data and model individual densities as warped versions of an underlying overall density. Unwarped distribution functions are obtained from an inverse warping mapping, and asymptotic properties of the synchronized overall density estimates are derived. Simulation results show that our method is advantageous over conventional density averaging. The approach complements previous quantile normalization methods used for microarray expression data and is illustrated with both longevity data of Mexican fruit flies and gene expression data of the Ts1Cje mouse study for Down syndrome.;We also consider functional analysis of recurrent events (FARE), where each subject for whom events are observed has an associated random local intensity function that corresponds to a realization of an underlying intensity process. We consider the two main components, the number of events and the distribution of events. Functional data analysis methodology is used to model and predict subject-specific densities by developing an estimate for the covariance structure of random density functions. We emphasize the case where for each density, there are only very few observations available. Functional principal component analysis is then applied to predict subject-specific density functions, from which one may also derive predictions of local intensity functions. Simulation results show that the proposed method outperforms conventional nonparametric density estimation methods and even parametric MLE estimates for densities. The effectiveness of FARE is demonstrated with mortality data from a Danish twin study, by fertility data on the timing of child births from a cohort of French-Canadian women, and with data on coughing and sputum events from a bronchial asthma treatment study. |
